# Null Ring iff Zero and Unity Coincide

## Theorem

The null ring is the only ring in which the unity and zero coincide.

## Proof

The single element of the null ring serves as an identity for both of the operations.

So, in this particular ring, the unity and the zero are the same element.

A non-null ring contains some non-zero element $a$.

Since $1_R \circ a = a \ne 0_R = 0_R \circ a$, then $1_R \ne 0_R$.

So if a ring is non-null, its unity cannot be zero.

So a ring is null if and only if its unity is also its zero.

$\blacksquare$